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Techniques, computations, and conjectures for semitopological Ktheory
 MATH. ANN
, 2004
"... We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence tha ..."
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Cited by 6 (1 self)
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We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic Ktheory of varieties, and it is also compatible with the classical AtiyahHirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the BorelMoore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semitopological Ktheory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational threefolds, and related varieties, the semitopological Kgroups and topological Kgroups are isomorphic in all degrees permitted by cohomological considerations. We also
Rational Isomorphisms between KTheories and Cohomology Theories
"... The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on ratio ..."
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Cited by 5 (2 self)
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The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishes a useful general criterion for a natural transformation of functors on quasiprojective complex varieties to induce a homotopy equivalence of semitopological singular complexes. Since semitopological Ktheory and morphic cohomology can be formulated as the semitopological singular complexes associated to Ktheory and motivic cohomology, this criterion provides a rational isomorphism between the semitopological Ktheory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a RiemannRoch theorem for the Chern character on semitopological Ktheory and an interpretation of the "topological filtration" on singular cohomology groups in K theoretic terms.
Semitopological Khomology and Thomason’s Theorem
, 2001
"... In this paper, we introduce the “semitopological Khomology” of complex varieties, a theory related to semitopological Ktheory much as connective topological Khomology is related to connective topological Ktheory. Our main theorem is that the semitopological Khomology of a smooth, quasiprojec ..."
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Cited by 4 (2 self)
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In this paper, we introduce the “semitopological Khomology” of complex varieties, a theory related to semitopological Ktheory much as connective topological Khomology is related to connective topological Ktheory. Our main theorem is that the semitopological Khomology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an. From this result, we deduce a pair of results relating semitopological Ktheory with connective topological Ktheory. In particular, we prove that the “Bott inverted ” semitopological Ktheory of a smooth, projective complex variety X coincides with the topological Ktheory of X an. In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason’s celebrated theorem that “Bott inverted” algebraic Ktheory with Z/n coefficients coincides with topological Ktheory with Z/n coefficients.
VANISHING THEOREMS FOR REAL ALGEBRAIC CYCLES
"... Abstract. We establish the analogue of the FriedlanderMazur conjecture for Teh’s reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasiprojective variety X vanishes in homological degrees larger than the dimension of X in all weights. As an ap ..."
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Abstract. We establish the analogue of the FriedlanderMazur conjecture for Teh’s reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasiprojective variety X vanishes in homological degrees larger than the dimension of X in all weights. As an application we obtain a vanishing of homotopy groups of the mod2 topological groups of averaged cycles and a characterization in a range of indices of dos Santos ’ real Lawson homology as the homotopy groups of the topological group of averaged cycles. We also establish an equivariant Poincare duality between equivariant FriedlanderWalker real morphic cohomology and dos Santos ’ real Lawson homology. We use this together with an equivariant extension of the mod2 BeilinsonLichtenbaum conjecture to compute some real
Münster J. of Math. 1 (2008), 99999–99999 Münster Journal of Mathematics c ○ Münster J. of Math. 2008 Ktheory of Leavitt path algebras
, 903
"... Abstract. Let E be a rowfinite quiver and let E0 be the set of vertices of E; consider the adjacency matrix N ′ E = (nij) ∈ Z (E0×E0), nij = # { arrows from i to j}. Write Nt E and 1 for the matrices ∈ Z(E0×E0\Sink(E)) which result from N ′t E and from the identity matrix after removing the columns ..."
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Abstract. Let E be a rowfinite quiver and let E0 be the set of vertices of E; consider the adjacency matrix N ′ E = (nij) ∈ Z (E0×E0), nij = # { arrows from i to j}. Write Nt E and 1 for the matrices ∈ Z(E0×E0\Sink(E)) which result from N ′t E and from the identity matrix after removing the columns corresponding to sinks. We consider the Ktheory of the Leavitt algebra LR(E) = LZ(E) ⊗ R. We show that if R is either a Noetherian regular ring or a stable C∗algebra, then there is an exact sequence (n ∈ Z)